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Properties

Label 16.0.18240547258...000.27
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 5^{12}\cdot 61^{12}$
Root discriminant $583.87$
Ramified primes $2, 5, 61$
Class number $6923321600$ (GRH)
Class group $[2, 2, 520, 3328520]$ (GRH)
Galois group $C_4^2$ (as 16T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2386255652001, 0, -152109439596, 0, 15664045662, 0, -630293848, 0, 27658415, 0, -777952, 0, 15082, 0, -164, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 164*x^14 + 15082*x^12 - 777952*x^10 + 27658415*x^8 - 630293848*x^6 + 15664045662*x^4 - 152109439596*x^2 + 2386255652001)
 
gp: K = bnfinit(x^16 - 164*x^14 + 15082*x^12 - 777952*x^10 + 27658415*x^8 - 630293848*x^6 + 15664045662*x^4 - 152109439596*x^2 + 2386255652001, 1)
 

Normalized defining polynomial

\( x^{16} - 164 x^{14} + 15082 x^{12} - 777952 x^{10} + 27658415 x^{8} - 630293848 x^{6} + 15664045662 x^{4} - 152109439596 x^{2} + 2386255652001 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(182405472588468433947695572320256000000000000=2^{48}\cdot 5^{12}\cdot 61^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $583.87$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 5, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4880=2^{4}\cdot 5\cdot 61\)
Dirichlet character group:    $\lbrace$$\chi_{4880}(1,·)$, $\chi_{4880}(2307,·)$, $\chi_{4880}(2561,·)$, $\chi_{4880}(843,·)$, $\chi_{4880}(2317,·)$, $\chi_{4880}(1231,·)$, $\chi_{4880}(853,·)$, $\chi_{4880}(599,·)$, $\chi_{4880}(4637,·)$, $\chi_{4880}(3427,·)$, $\chi_{4880}(3173,·)$, $\chi_{4880}(3049,·)$, $\chi_{4880}(1963,·)$, $\chi_{4880}(1719,·)$, $\chi_{4880}(111,·)$, $\chi_{4880}(489,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{9} a^{6} + \frac{1}{9} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{27} a^{7} + \frac{4}{27} a^{5} + \frac{4}{27} a^{3} - \frac{1}{3} a$, $\frac{1}{1080} a^{8} - \frac{1}{54} a^{7} - \frac{1}{540} a^{6} - \frac{2}{27} a^{5} - \frac{101}{1080} a^{4} - \frac{2}{27} a^{3} + \frac{17}{180} a^{2} + \frac{1}{6} a + \frac{13}{40}$, $\frac{1}{3240} a^{9} - \frac{7}{540} a^{7} - \frac{29}{360} a^{5} - \frac{1}{6} a^{4} - \frac{29}{1620} a^{3} + \frac{1}{6} a^{2} - \frac{41}{360} a - \frac{1}{2}$, $\frac{1}{16200} a^{10} - \frac{23}{1080} a^{6} - \frac{1}{6} a^{5} + \frac{109}{810} a^{4} - \frac{1}{6} a^{3} + \frac{29}{120} a^{2} - \frac{1}{6} a + \frac{11}{100}$, $\frac{1}{210600} a^{11} + \frac{1}{21060} a^{9} - \frac{19}{1560} a^{7} - \frac{1}{18} a^{6} - \frac{2383}{21060} a^{5} - \frac{1}{18} a^{4} + \frac{3547}{42120} a^{3} - \frac{7}{18} a^{2} + \frac{1147}{5850} a$, $\frac{1}{6696448200} a^{12} - \frac{12014}{837056025} a^{10} + \frac{51517}{446429880} a^{8} - \frac{1}{54} a^{7} - \frac{5621969}{334822410} a^{6} + \frac{5}{54} a^{5} + \frac{56877611}{1339289640} a^{4} + \frac{5}{54} a^{3} - \frac{86104961}{372024900} a^{2} + \frac{1}{3} a - \frac{35423}{176650}$, $\frac{1}{6696448200} a^{13} - \frac{721}{6696448200} a^{11} - \frac{5669}{111607470} a^{9} - \frac{4458977}{1339289640} a^{7} - \frac{45727697}{669644820} a^{5} - \frac{36234749}{248016600} a^{3} - \frac{41206507}{82672200} a$, $\frac{1}{3657248081701297200} a^{14} - \frac{1}{13392896400} a^{13} + \frac{122525557}{1828624040850648600} a^{12} + \frac{721}{13392896400} a^{11} + \frac{464147009861}{81272179593362160} a^{10} + \frac{5669}{223214940} a^{9} + \frac{65556563201399}{146289923268051888} a^{8} - \frac{45144343}{2678579280} a^{7} - \frac{22894190686219093}{731449616340259440} a^{6} - \frac{53478943}{1339289640} a^{5} + \frac{12872349378766289}{1219082693900432400} a^{4} - \frac{508451}{496033200} a^{3} + \frac{5662876257389903}{16931704081950450} a^{2} - \frac{13908293}{165344400} a - \frac{2508385551389}{5937042851440}$, $\frac{1}{16095548807567408977200} a^{15} - \frac{196641103169}{16095548807567408977200} a^{13} - \frac{1}{13392896400} a^{12} - \frac{5692724471862721}{8047774403783704488600} a^{11} - \frac{317249}{13392896400} a^{10} - \frac{21019118781974017}{643821952302696359088} a^{9} + \frac{90461}{223214940} a^{8} + \frac{11819304187162350679}{804777440378370448860} a^{7} + \frac{48529619}{2678579280} a^{6} - \frac{1332415130996073247603}{16095548807567408977200} a^{5} - \frac{36235139}{267857928} a^{4} - \frac{115444319477203612037}{1788394311951934330800} a^{3} + \frac{62669257}{1488099600} a^{2} + \frac{29586222639491821}{212297520412147950} a + \frac{293611}{1413200}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}\times C_{520}\times C_{3328520}$, which has order $6923321600$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 773433248.8138483 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2$ (as 16T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4^2$
Character table for $C_4^2$

Intermediate fields

\(\Q(\sqrt{61}) \), \(\Q(\sqrt{610}) \), \(\Q(\sqrt{10}) \), 4.4.363169600.2, \(\Q(\sqrt{10}, \sqrt{61})\), 4.4.3631696.1, 4.0.58107136000.4, 4.0.58107136000.7, 4.0.256000.2, 4.0.952576000.2, 8.8.2110274533826560000.1, 8.0.3376439254122496000000.5, 8.0.907401035776000000.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.1.0.1}{1} }^{16}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
61Data not computed